This may be a very broad question for which I am sorry. But this may be thought of as a survey of existing methods (at least some useful methods).
Suppose $X_n$ is a sequence of random variables adapted to the filtration $\mathcal F_n$. Suppose we can write $X_n=A_n+D_n$ where $A_n=E(X_n|\mathcal F_{n-1})$ and $D_n=X_n-E(X_n|\mathcal F_{n-1})$.
When can we say that $X_n$ converges? That is, what are useful conditions on $A_n,D_n$ such that we can say $X_n$ converges?
I know Doob's convergence theorem related to sub- and super-martingales. Also I am aware of the result that a uniformly integrable martingale converges.